相关论文: Growth of root multiplicities along imaginary root…

We present combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from the realization of the corresponding infinity-crystal using quiver varieties. The framework is general,…

We name an indecomposable symmetrizable generalized Cartan matrix $A$ and the corresponding Kac--Moody Lie algebra ${\goth g} ^\prime (A)$ {\it of the arithmetic type} if for any $\beta \in Q$ with $(\beta | \beta)<0$ there exist $n(\beta…

In this paper we consider those involutions $\theta$ of a finite-dimensional Kac-Moody Lie superalgebra $\mathfrak g$, with associated decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$, for which a Cartan subspace $\mathfrak a$ in…

We discuss the known results and methods for determining root multiplicities for hyperbolic Kac--Moody algebras.

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of…

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the…

In this paper we extend several results about root systems of Kac-Moody algebras to superalgebra context. In particular, we describe the root bases and the sets of imaginary roots.

Root multiplicities encode information about the structure of Kac-Moody algebras, and appear in applications as far-reaching as string theory and the theory of modular functions. We provide an algorithm based on the Peterson recurrence…

In this paper, we calculate the dimension of root spaces $\mathfrak{g}_{\lambda}$ of a special type rank $3$ Kac-Moody algebras $\mathfrak{g}$. We first introduce a special type of elements in $\mathfrak{g}$, which we call elements in…

Let k be a field and A the n-Kronecker algebra, this is the path algebra of the quiver with 2 vertices, a source and a sink, and n arrows from the source to the sink. It is well-known that the dimension vectors of the indecomposable…

Let $G$ be a finite group with symmetric generating set $S$, and let $c = \max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding Cayley graph, where $B(R)$ denotes an $R$-ball in the word-metric with respect to $S$. We…

We study the growth rate of some power-free languages. For any integer $k$ and real $\beta>1$, we let $\alpha(k,\beta)$ be the growth rate of the number of $\beta$-free words of a given length over the alphabet $\{1,2,\ldots, k\}$. Shur…

In this note we explicitly construct top-dimensional components of the cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands $V(\lambda+\mu-N\beta) \subset V(\lambda)…

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number $\beta$. We prove that if $\beta$ is real quadratic, then the number of partitions is always finite if and only if some conjugate of $\beta$…

Consider any nonzero univariate polynomial with rational coefficients, presented as an elementary algebraic expression (using only integer exponents). Letting sigma(f) denotes the additive complexity of f, we show that the number of…

Let $G$ be a graph with an adjacent matrix $A(G)$. The multiplicity of an arbitrary eigenvalue $\lambda$ of $A(G)$ is denoted by $m_\lambda(G)$. In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of $T$…

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of…

The Fundamental Theorem of Algebra (FTA) asserts that every complex polynomial has as many complex roots, counted with multiplicities, as its degree. A probabilistic analogue of this theorem for real roots of real polynomials, sometimes…

The derived algebra of a symmetrizible Kac-Moody algebra $\lie g$ is generated (as a Lie algebra) by its root spaces corresponding to real roots. In this paper, we address the natural reverse question: given any subset of real root vectors,…

For each $\alpha \in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{\alpha +…